Linear Dynamic Analysis using Shear Deformable Frame Element
- Adisorn O.
- 2 hours ago
- 4 min read
Adisorn Owatsiriwong, D.Eng.
ALPS Consultants
1. Introduction
Analysis of seismic-resistant frame structures might demand considering shear deformation (Bischoff, 2002; Bathe, 1996) ie., simplified wall/pier elements in global frame analysis. Shear-deformable frame elements are also useful for modeling members where shear deformation is not negligible, such as deep beams, short columns, coupling beams.
Integrating the FEM with implicit time integration is crucial to handling time-varying ground acceleration data. For a slender frame structure, the P-Delta effect must be included in the analysis, and it is much more important for a frame exhibiting plastic behavior. As the first part of our project, we developed a FEM code on MATLAB for linear dynamic analysis of a 2D frame using a shear deformable frame and the Newmark implicit integration scheme. The program can show time-history plots of nodal displacement, velocity, and acceleration, as well as the internal forces developed at each time step.
2. Shear Deformable Beam Formulation
2.1 Strain-displacement relationship
The displacement field variable at any point is
Curvature is simply
, and shear strain is defined as
(2.1)
Note:For Bernoulli beam,
2.2 Constitutive equation
(2.2)
2.3 Equilibrium equations
The equilibrium equation can be formulated either by Galerkin's weighted residual method or, equivalently, by energy methods. In this study, we use the principle of virtual work, i.e.
(2.3)
Substituting (2.1) and (2.2) into (2.3) gives
(2.4)
3. Discretization and Element Stiffness
(2.5)
where
is the shape function in natural coordinates
are nodal variables.
The strain-displacement matrix relating field variable and the nodal variable can be written as
Replace the field variables with their discretized version into (2.4) and we can write the element-level equilibrium equation in matrix form, i.e.
where
is the member's end-foce vector
4. Extension to Frame Element in 2D
The element stiffness can be simply extended to frame element by adding axial translation dof and the corresponding axial stiffness
The nodal variable is coupled by the shear stiffness matrix, i.e., is a full vector, while axial stiffness is uncoupled as a standard frame formulation. is equivalent nodal force vector due to member's force, i.e., distributed loads, prestressing, initial strain, etc.
The transformation matrix in 2D plane is written as
where
and for nodal displacement, the transformation matrix can be written as
Then the stiffness matrix can be transformed into the global coordinates using
, for example and the force vector in the global coordinates is
Then the system equation can be obtained for each node, i.e.
5. P-Delta Effect and Iteration
For a frame with lateral displacement, the second-order moment occurs, i.e.
. Then the virtual external work due to this second-order moment is
, by applying integration by part
This causing by element axial force and the displacement is not part of strain energy but the external force.
or in a discretized form
Now the equilibrium equation can be rewritten as
at any node, or
for the complete structure
The solution involving P-Delta effect requires Newton iteration. The residual of force equilibrium is written as
where r_i is the imbalance force at load step i
Then we can write the incremental equilibrium equation as
and
It's noted that the geometric has destabilization effect to the system stiffness. It should be stressed that only are the member forces using for stress calcualtion, i.e. for nonlinear material problems. shall be treated as part of external nodal force, but adjusting by the displacement where Newton iteration loop is applied.
6. Newmark Method for Direct Time Integration
The dynamic equilibrium equation of the structure considering seismic excitation and P-Delta load effect at time t+1 can be written by
(6.1)
where are mass and friction or damping matrix. Those matrices are usually diagonal for efficient processing by reduction technique (Guyan, 1965; Bathe, 1996; Paz, 2004)
Applying Newmark integration sceme with
turns (Oechsner, 2021) into a simple first order equation in terms of
displacement , i.e.,
where
The completed derivation can be found in major textbooks (Bathe, 1996; Oesner, 2021 ) for example.
7. Numerical Example
A 5-story rigid framed is subjected to ground accerelation at base (Fig. 5.1). The properties of material used for beam and columns are listed as below
E = 25e6; % kN/m2
G = 10e6; % kN/m2
kappa = 5/6;
rho = 25/9.81; % kN*s2/m4
% Set 1: column
b = 0.40; h = 0.40;
A = b*h;
I = b*h^3/12;
propSet(1).name = 'Column';
propSet(1).EA = E*A;
propSet(1).EI = E*I;
propSet(1).kGA = kappa*G*A;
propSet(1).rhoA = rho*A;
% Set 2: beam
b = 0.30; h = 0.50;
A = b*h;
I = b*h^3/12;
propSet(2).name = 'Beam';
propSet(2).EA = E*A;
propSet(2).EI = E*I;
propSet(2).kGA = kappa*G*A;
propSet(2).rhoA = rho*A;
Fig. 7.1 Ground accerelation data
Fig. 7.2 Deformed shape
Fig. 7.3 Analysis results for roof displacement, velocity and acceleration
8. Conclusion
This study formulated a 2D shear-deformable Timoshenko frame element including axial, bending, shear, transformation, mass, damping, and geometric stiffness terms.
The MATLAB implementation demonstrated static and Newmark seismic dynamic analysis with user-defined section properties for beams and columns. The next phase of the study shall include nonlinear material analysis, especially lumped plastic model.
9. References
M Bischoff, Advanced Finite Element Method, Lecture notes at Technische Universität München, 2002
KJ Bathe, Finite Element Procedures, Prentice Hall, 1996
Mario Paz and William Leigh, Structural Dynamics: Theory and Computation, 5th Edition, Springer, 2004
A Oechsner, Computational Statics and Dynamics, Springer, 2021